Serialized PHP data is just a string, so for "simple" searches, you can use basic MySQL string operations.
e.g. The serialized array has a name
parameter and you need to match against that, so
... WHERE LOCATE(CONCAT('s:', LENGTH('foo'), ':foo') ,serializeddatafield) > 0
which would produce
WHERE LOCATE('s:3:foo', serializeddata) > 0
But then, this would find ANY strings whose value is foo
anywhere in the data, and not necessarily in the particular array field you want.
e.g. you're probably better off yanking out the whole field, deserializing in PHP, and then doing the search there. You could use the above technique to minimize the total number of strings you'd have to yank/parse, however.